Added May 31, 2019.
Problem Chapter 6.8.1.1, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + f(x) w_y + g(x) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + f[x]*D[w[x, y,z], y] +g[x]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y-\int _1^xf(K[1])dK[1],z-\int _1^xg(K[2])dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ f(x)*diff(w(x,y,z),y)+g(x)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y -\left (\int f \left (x \right )d x \right ), z -\left (\int g \left (x \right )d x \right )\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.1.2, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + f(x) (y+a) w_y + g(x) (z+b) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + f[x]*(y+a)*D[w[x, y,z], y] +g[x]*(z+b)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^xf(K[1])dK[1]\right )-\int _1^xa \exp \left (-\int _1^{K[2]}f(K[1])dK[1]\right ) f(K[2])dK[2],z \exp \left (-\int _1^xg(K[3])dK[3]\right )-\int _1^xb \exp \left (-\int _1^{K[4]}g(K[3])dK[3]\right ) g(K[4])dK[4]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ f(x)*(y+a)*diff(w(x,y,z),y)+g(x)*(z+b)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\left (a +y \right ) {\mathrm e}^{-\left (\int f \left (x \right )d x \right )}, \left (b +z \right ) {\mathrm e}^{-\left (\int g \left (x \right )d x \right )}\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.1.3, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (a y + f(x)) w_y + (b z+g(x)) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (a*y+f[x])*D[w[x, y,z], y] +(b*z+g[x])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y e^{-a x}-\int _1^xe^{-a K[1]} f(K[1])dK[1],z e^{-b x}-\int _1^xe^{-b K[2]} g(K[2])dK[2]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (a*y+f(x))*diff(w(x,y,z),y)+(b*z+g(x))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-a x}-\left (\int {\mathrm e}^{-a x} f \left (x \right )d x \right ), z \,{\mathrm e}^{-b x}-\left (\int {\mathrm e}^{-b x} g \left (x \right )d x \right )\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.1.4, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y + f_2(x)) w_y + (g_1(x) y+g_2(x)) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] +(g1[x]*y+g2[x])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[3]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[3])dK[3],-\int _1^x\left (\text {g2}(K[4])-\exp \left (-\int _1^{K[4]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[4]) \int _1^{K[4]}\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]\right )dK[4]-y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right ) \int _1^x\exp \left (\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {g1}(K[2])dK[2]+z\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+(g1(x)*y+g2(x))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z -\left (\int _{}^{x}\left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f} \right )} \mathit {f2} \left (\mathit {\_f} \right )d \mathit {\_f} \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ) {\mathrm e}^{\int \mathit {f1} \left (\mathit {\_f} \right )d \mathit {\_f}} \mathit {g1} \left (\mathit {\_f} \right )+\mathit {g2} \left (\mathit {\_f} \right )\right )d\mathit {\_f} \right )\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.1.5, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y + f_2(x)) w_y + (g_1(x) z+g_2(x)) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x])*D[w[x, y,z], y] +(g1[x]*z+g2[x])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (y \exp \left (-\int _1^x\text {f1}(K[1])dK[1]\right )-\int _1^x\exp \left (-\int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2],z \exp \left (-\int _1^x\text {g1}(K[3])dK[3]\right )-\int _1^x\exp \left (-\int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x))*diff(w(x,y,z),y)+(g1(x)*z+g2(x))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y \,{\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z \,{\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )}-\left (\int {\mathrm e}^{-\left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right )\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.1.6, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_2(x) y + f_1(x) z+f_0(x)) w_y + (g_2(x) y+g_1(x) z + g_0(x)) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f2[x]*y+f1[x]*z+f0[x])*D[w[x, y,z], y] +(g2[x]*y+g1[x]*z+g0[x])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✗
restart; pde := diff(w(x,y,z),x)+ (f2(x)*y+f1(x)*z+f0(x))*diff(w(x,y,z),y)+(g2(x)*y+g1(x)*z+g0(x))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
sol=()
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Added May 31, 2019.
Problem Chapter 6.8.1.7, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (y^2-a^2+a \lambda \sinh (\lambda x)-a^2 \sinh ^2(\lambda x)) w_y + f(x) \sinh (\gamma z) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (y^2-a^2+a*lambda*Sinh[lambda*x]-a^2*Sinh[lambda*x]^2)*D[w[x, y,z], y] +f[x]*Sinh[gamma*z]*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left (\frac {\log \left (\tanh \left (\frac {\gamma z}{2}\right )\right )}{\gamma }-\int _1^xf(K[2])dK[2],\frac {2 \lambda e^{\frac {a e^{-\lambda x} \left (e^{2 \lambda x}-1\right )}{\lambda }+\lambda x}}{a e^{2 \lambda x}+a-2 y e^{\lambda x}}-\int _1^{e^{\lambda x}}\frac {e^{\frac {a \left (K[1]^2-1\right )}{\lambda K[1]}}}{K[1]}dK[1]\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (y^2-a^2+a*lambda*sinh(lambda*x)-a^2*sinh(lambda*x)^2)*diff(w(x,y,z),y)+f(x)*sinh(gamma*z)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (-\frac {2 \sqrt {\sinh \left (\lambda x \right )+i}\, \left (-\left (-\frac {\left (\sinh ^{2}\left (\lambda x \right )\right )}{2}+i \sinh \left (\lambda x \right )+\frac {1}{2}\right ) \lambda \HeunCPrime \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right )+\left (i \left (\sinh ^{2}\left (\lambda x \right )\right )+2 \sinh \left (\lambda x \right )-i\right ) \left (a \cosh \left (\lambda x \right )+y \right ) \HeunC \left (\frac {4 i a}{\lambda }, -\frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )\right )}{-\left (-\sinh \left (\lambda x \right )+i\right ) \left (\sinh ^{2}\left (\lambda x \right )+1\right ) \lambda \HeunCPrime \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right ) \cosh \left (\lambda x \right )+\left (2 \left (i \sinh \left (\lambda x \right )+1\right ) \left (\sinh ^{2}\left (\lambda x \right )+1\right ) y +\left (2 i a \left (\sinh ^{3}\left (\lambda x \right )\right )+\left (i \lambda +2 a \right ) \left (\sinh ^{2}\left (\lambda x \right )\right )+2 a -i \lambda +\left (2 i a +2 \lambda \right ) \sinh \left (\lambda x \right )\right ) \cosh \left (\lambda x \right )\right ) \HeunC \left (\frac {4 i a}{\lambda }, \frac {1}{2}, -\frac {1}{2}, \frac {2 i a}{\lambda }, -\frac {8 i a -3 \lambda }{8 \lambda }, -\frac {i \sinh \left (\lambda x \right )}{2}+\frac {1}{2}\right )}, \frac {-2 \arctanh \left ({\mathrm e}^{\gamma z}\right )-\gamma \left (\int f \left (x \right )d x \right )}{\gamma }\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.1.8, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x) z+g_2(x) z^m) w_z = 0 \]
Mathematica ✓
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] +(g1[x]*z+g2[x]*z^m)*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
\[\left \{\left \{w(x,y,z)\to c_1\left ((k-1) \int _1^x\exp \left ((k-1) \int _1^{K[2]}\text {f1}(K[1])dK[1]\right ) \text {f2}(K[2])dK[2]+y^{1-k} \exp \left ((k-1) \int _1^x\text {f1}(K[1])dK[1]\right ),(m-1) \int _1^x\exp \left ((m-1) \int _1^{K[4]}\text {g1}(K[3])dK[3]\right ) \text {g2}(K[4])dK[4]+z^{1-m} \exp \left ((m-1) \int _1^x\text {g1}(K[3])dK[3]\right )\right )\right \}\right \}\]
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+(g1(x)*z+g2(x)*z^m)*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), z^{-m +1} {\mathrm e}^{\left (m -1\right ) \left (\int \mathit {g1} \left (x \right )d x \right )}+\left (m -1\right ) \left (\int {\mathrm e}^{\left (m -1\right ) \left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right )\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.1.9, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x) y+f_2(x) y^k) w_y + (g_1(x)+g_2(x) e^{\lambda z}) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]*y+f2[x]*y^k)*D[w[x, y,z], y] +(g1[x]+g2[x]*Exp[lambda*z])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (f1(x)*y+f2(x)*y^k)*diff(w(x,y,z),y)+(g1(x)+g2(x)*exp(lambda*z))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (y^{-k +1} {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )}+\left (k -1\right ) \left (\int {\mathrm e}^{\left (k -1\right ) \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right ), \frac {-\lambda \left (\int {\mathrm e}^{\lambda \left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right )-{\mathrm e}^{-\left (z -\left (\int \mathit {g1} \left (x \right )d x \right )\right ) \lambda }}{\lambda }\right )\]
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Added May 31, 2019.
Problem Chapter 6.8.1.10, from Handbook of first order partial differential equations by Polyanin, Zaitsev, Moussiaux.
Solve for \(w(x,y,z)\)
\[ w_x + (f_1(x)+f_2(x) e^{\lambda y}) w_y + (g_1(x)+g_2(x) e^{\beta z}) w_z = 0 \]
Mathematica ✗
ClearAll["Global`*"]; pde = D[w[x, y,z], x] + (f1[x]+f2[x]*Exp[lambda*y])*D[w[x, y,z], y] +(g1[x]+g2[x]*Exp[beta*z])*D[w[x,y,z],z]==0; sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, y,z], {x, y,z}], 60*10]];
Failed
Maple ✓
restart; pde := diff(w(x,y,z),x)+ (f1(x)+f2(x)*exp(lambda*y))*diff(w(x,y,z),y)+(g1(x)+g2(x)*exp(beta*z))*diff(w(x,y,z),z)= 0; cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,y,z))),output='realtime'));
\[w \left (x , y , z\right ) = \mathit {\_F1} \left (\frac {-\lambda \left (\int {\mathrm e}^{\lambda \left (\int \mathit {f1} \left (x \right )d x \right )} \mathit {f2} \left (x \right )d x \right )-{\mathrm e}^{-\left (y -\left (\int \mathit {f1} \left (x \right )d x \right )\right ) \lambda }}{\lambda }, \frac {-\beta \left (\int {\mathrm e}^{\beta \left (\int \mathit {g1} \left (x \right )d x \right )} \mathit {g2} \left (x \right )d x \right )-{\mathrm e}^{-\left (z -\left (\int \mathit {g1} \left (x \right )d x \right )\right ) \beta }}{\beta }\right )\]
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